Journal of Symbolic Logic

Hybrid Logics: Characterization, Interpolation and Complexity

Carlos Areces, Patrick Blackburn, and Maarten Marx

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Hybrid languages are expansions of propositional modal languages which can refer to (or even quantify over) worlds. The use of strong hybrid languages dates back to at least [Pri67], but recent work (for example [BS98, BT98a, BT99]) has focussed on a more constrained system called $\mathscr{H}(\downarrow, @)$. We show in detail that $\mathscr{H}(\downarrow, @)$ is modally natural. We begin by studying its expressivity, and provide model theoretic characterizations (via a restricted notion of Ehrenfeucht-Fraisse game, and an enriched notion of bisimulation) and a syntactic characterization (in terms of bounded formulas). The key result to emerge is that $\mathscr{H}(\downarrow, @)$ corresponds to the fragment of first-order logic which is invariant for generated submodels. We then show that $\mathscr{H}(\downarrow, @)$ enjoys (strong) interpolation, provide counterexamples for its finite variable fragments, and show that weak interpolation holds for the sublanguage $\mathscr{H}$(@). Finally, we provide complexity results for $\mathscr{H}$(@) and other fragments and variants, and sharpen known undecidability results for $\mathscr{H}(\downarrow, @)$.

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J. Symbolic Logic, Volume 66, Issue 3 (2001), 977-1010.

First available in Project Euclid: 6 July 2007

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Areces, Carlos; Blackburn, Patrick; Marx, Maarten. Hybrid Logics: Characterization, Interpolation and Complexity. J. Symbolic Logic 66 (2001), no. 3, 977--1010.

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